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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmcoss | Structured version Visualization version GIF version |
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.) |
Ref | Expression |
---|---|
eldmcoss | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmcoss3 38151 | . . 3 ⊢ dom ≀ 𝑅 = dom ◡𝑅 | |
2 | 1 | eleq2i 2818 | . 2 ⊢ (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ∈ dom ◡𝑅) |
3 | eldmcnv 38043 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | |
4 | 2, 3 | bitrid 282 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wex 1774 ∈ wcel 2099 class class class wbr 5153 ◡ccnv 5681 dom cdm 5682 ≀ ccoss 37876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-coss 38109 |
This theorem is referenced by: eldmcoss2 38157 eldm1cossres 38158 |
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